nathanl6656

98 Questions
19 Answers

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Algebra

Suppose the polynomial p(x)=x^3+ax^2+bc+c has the property that the mean of its zeroes, the product of its zeroes, and the sum of its coefficients are all equal. If the y-intercept of the graph of y=p(x) is 0, what is b?

nathanl6656 02.03.2025

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4

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Algebra

Find all (real or nonreal) x satisfying
(x - 3)^4 + (x - 5)^4 = -8 + 6(x - 3)(x - 5)^3 - 11(x - 3)^3 (x - 5).

nathanl6656 02.03.2025

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3

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Algebra

Express x^8 + x^4 y^4 - x^6 y^2 + x^3 y^5 - 4xy^7 - y^8 as the product of two polynomials of degree 2, and a polynomial of degree 4.

nathanl6656 02.03.2025

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3

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Algebra

Find the constant k so that
2x^2 + 5xy - 8y^2 + 7x + 25y + k
can be expressed as the product of two linear factors of the form ax + by + c.

nathanl6656 02.03.2025

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4

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Algebra

Solve the system of equations
x + y + 2z = 1
-2x - 4y + 2z = 7
10x - 17x + 13z = 8 mehr ..

nathanl6656 22.02.2025

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6

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Algebra

Solve the system of equations
5x + 3z = 1
-x + y - z = 0
2x - 3y + 7z = -4

nathanl6656 22.02.2025

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Algebra

Given that
5a - 7b + 3c = 19
4a + 2b - 8c = 4
7a + 5b + 3c = -11 mehr ..

nathanl6656 22.02.2025

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4

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Algebra

Find all values of z such that z^4 - 4z^2 + 3 = 5z^2 + 11z^3 - 40.

nathanl6656 22.02.2025

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12

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Counting

At a meeting, $5$ scientists, $2$ mathematicians, and a journalist are to be seated around a circular table. How many different arrangements are possible if each scientist must sit next to a mathematician? (Two seatings are considered equivalent mehr ..

nathanl6656 23.01.2025

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12

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Counting

Four children and four adults are to be seated at a circular table. In how many different ways can they be seated if all the children are next to each other, and all the adults are next to each other? (Two seatings are considered the same if mehr ..

nathanl6656 23.01.2025

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9

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Counting

I have $4$ different mathematics textbooks and $3$ different psychology textbooks. In how many ways can I place the $7$ textbooks on a bookshelf, in a row, if there must be a mathematics textbook exactly in the middle, and all the psychology textbooks are mehr ..

nathanl6656 23.01.2025

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11

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Algebra

A line bisects the area of the square below. The slope of the line is 3, and its y-intercept is (0,b). Find $b.$
The vertices of the square are (-4,-4), (-4,4), (4,4), and (4,-4).

nathanl6656 01.01.2025

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12

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Algebra

Find the coordinates of the intersection of the diagonals of quadrilateral ABCD, where:
A(3, 6), B(9, 6), C(7, 3), and D(1, 1).

nathanl6656 01.01.2025

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16

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+413

Algebra

The graph x^2+y^2=121 intersects y=8x+10 at two points P and Q. Find PQ.

nathanl6656 01.01.2025

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15

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Algebra

The line 3x-4y=5 intersects the $x$-axis and $y$-axis at points $A$ and $B,$ respectively. Find the distance $AB.$

nathanl6656 01.01.2025

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20

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Geometry

Two tangents $\overline{PA}$ and $\overline{PB}$ are drawn to a circle, where $P$ lies outside the circle, and $A$ and $B$ lie on the circle. The length of $\overline{AB}$ is $4,$ and the circle has a radius of $5.$ Find the length $AB.$ mehr ..

nathanl6656 13.12.2024

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20

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Geometry

Lines $XQ$ and $XR$ are tangent to a circle, as shown below. If $\angle XTU = 47^\circ$ and $\angle QAU = 65^\circ,$ then find $\angle QXR,$ in degrees.

nathanl6656 13.12.2024

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17

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Geometry

A circular table is pushed into a corner of the room, where two walls meet at a right angle. A point P on the edge of the table (as shown below) has a distance of 6 from one wall, and a distance of 6 from the other wall. Find the radius of the mehr ..

nathanl6656 13.12.2024